| Hopf π-coalgebra was first used by V.G. Turaev, its original algebraic structure was extended from the single space to a family of space. Further, A. Vielizier generalized some algebraic properties of Hopf π-coalgebra.In this paper, we mainly consider the categories relationships for right C1-comodules category, right group-C-comodules category and right π-C x (?) kπ-comodules category. Be-side, We show the isomorphism conditions between π-coalgebra and7r-cross coproduct.The thesis is mainly made up of three parts as follows:The first chapter, We give the basic definitions and notions used in this paper, such as π-coalgebras, π-comodules, group-comodules, etc.The second chapter, we define strong π-coalgebra, and study its properties (theorem2.2). Furthermore, we give the properties of π-comodules and π-comodules morphism (corollary2.1, corollary2.2).Then, we discuss the categories relationships of right C1-comodule categories, right group-C-comodule categories, right π-C x (?) kπ-comodule categories.To begin with, cotensor product was used to construct π-coalgebra. Then, we show there exist categories equivalence between MC1and ΜπC when C is strong π-coalgebra.Next, we show the properties of group-C (?) kπ (lemma2.3). After that, it was proved there exist categories isomorphism between ΜπC and ΜC(?)kπ (theorem2.4).In the third place, based on the results obtained, we show category MCl is equivalent to category ΜC(?)kπ (theorem2.5).Last, we define π-cross coproduct, and give its equivalent conditions (theorem3.1). Further, it was committed to construct a π-cross coproduct which was isomorphic to the given π-cross coproduct.The third chapter, we show the equivalent conditions of categories isomorphism between π-coalgebra and π-cross coproduct (theorem3.2). As a special case of π-cross coproduct, we also give corresponding equivalent conditions of categories isomorphism between π-coalgebra and π-smash coproduct (theorem3.4). |
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